Multivariable calculus

I am currently reading baby Rudin, but I only know single-variable calculus at the moment, so I think it would be a good idea to learn the multi-variable stuff non-rigorously before I do the analysis in Rudin (chapters 9-11).

So I was thinking of either getting one of the many 'Mathematical methods for...' books or 'Calculus vol.2' by Apostol. Which would be better?

Why not a cheap book like the following one? I'm thinking, why not finish with the single-variable topics first, get them done and out the way, then learn the multivariable material rigorously from the start? I mean, you're learning analysis so you might as well put it to use.

I am currently reading baby Rudin, but I only know single-variable calculus at the moment, so I think it would be a good idea to learn the multi-variable stuff non-rigorously before I do the analysis in Rudin (chapters 9-11).

So I was thinking of either getting one of the many 'Mathematical methods for...' books or 'Calculus vol.2' by Apostol. Which would be better?

Apostol is pretty rigorous and it will take you quite a while to hack your way through. If you want a less rigorous but really excellent introduction, get Lang's Calculus of Several Variables.

Neither Apostol nor Lang does differential forms, though, and Rudin would be a horrible place to learn this (or any of the material in chapters 9-11, for that matter). An alternative would be Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, which I have not read but which has a very good reputation. Personally I don't see the point of trying to learn differential forms without having learned the "classical" treatment as in Lang, but that's just my preference.

If I recall correctly, Rudin does measure theory and Lebesgue integration in chapter 10 and/or 11. None of the above books will help you with this, and I would NOT advise learning it from Rudin. Almost any other book covering this material will be a better choice. A nice efficient (but expensive) choice would be Bartle's The Elements of Integration and Lebesgue Measure.